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Woo JH, Honey CJ, Moon JY. Phase and amplitude dynamics of coupled oscillator systems on complex networks. CHAOS (WOODBURY, N.Y.) 2020; 30:121102. [PMID: 33380037 PMCID: PMC7714526 DOI: 10.1063/5.0031031] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/27/2020] [Accepted: 11/05/2020] [Indexed: 06/12/2023]
Abstract
We investigated locking behaviors of coupled limit-cycle oscillators with phase and amplitude dynamics. We focused on how the dynamics are affected by inhomogeneous coupling strength and by angular and radial shifts in coupling functions. We performed mean-field analyses of oscillator systems with inhomogeneous coupling strength, testing Gaussian, power-law, and brain-like degree distributions. Even for oscillators with identical intrinsic frequencies and intrinsic amplitudes, we found that the coupling strength distribution and the coupling function generated a wide repertoire of phase and amplitude dynamics. These included fully and partially locked states in which high-degree or low-degree nodes would phase-lead the network. The mean-field analytical findings were confirmed via numerical simulations. The results suggest that, in oscillator systems in which individual nodes can independently vary their amplitude over time, qualitatively different dynamics can be produced via shifts in the coupling strength distribution and the coupling form. Of particular relevance to information flows in oscillator networks, changes in the non-specific drive to individual nodes can make high-degree nodes phase-lag or phase-lead the rest of the network.
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2
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Nguimdo RM. Constructing Hopf bifurcation lines for the stability of nonlinear systems with two time delays. Phys Rev E 2018; 97:032211. [PMID: 29776163 DOI: 10.1103/physreve.97.032211] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/30/2017] [Indexed: 11/07/2022]
Abstract
Although the plethora real-life systems modeled by nonlinear systems with two independent time delays, the algebraic expressions for determining the stability of their fixed points remain the Achilles' heel. Typically, the approach for studying the stability of delay systems consists in finding the bifurcation lines separating the stable and unstable parameter regions. This work deals with the parametric construction of algebraic expressions and their use for the determination of the stability boundaries of fixed points in nonlinear systems with two independent time delays. In particular, we concentrate on the cases for which the stability of the fixed points can be ascertained from a characteristic equation corresponding to that of scalar two-delay differential equations, one-component dual-delay feedback, or nonscalar differential equations with two delays for which the characteristic equation for the stability analysis can be reduced to that of a scalar case. Then, we apply our obtained algebraic expressions to identify either the parameter regions of stable microwaves generated by dual-delay optoelectronic oscillators or the regions of amplitude death in identical coupled oscillators.
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Affiliation(s)
- Romain Modeste Nguimdo
- Optique Nonlinéaire Théorique, Université Libre de Bruxelles, Campus Plaine, CP 231, 1050 Bruxelles, Belgium
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Mirasso CR, Carelli PV, Pereira T, Matias FS, Copelli M. Anticipated and zero-lag synchronization in motifs of delay-coupled systems. CHAOS (WOODBURY, N.Y.) 2017; 27:114305. [PMID: 29195321 DOI: 10.1063/1.5006932] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Anticipated and zero-lag synchronization have been observed in different scientific fields. In the brain, they might play a fundamental role in information processing, temporal coding and spatial attention. Recent numerical work on anticipated and zero-lag synchronization studied the role of delays. However, an analytical understanding of the conditions for these phenomena remains elusive. In this paper, we study both phenomena in systems with small delays. By performing a phase reduction and studying phase locked solutions, we uncover the functional relation between the delay, excitation and inhibition for the onset of anticipated synchronization in a sender-receiver-interneuron motif. In the case of zero-lag synchronization in a chain motif, we determine the stability conditions. These analytical solutions provide an excellent prediction of the phase-locked regimes of Hodgkin-Huxley models and Roessler oscillators.
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Affiliation(s)
- Claudio R Mirasso
- Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), Universitat de les Illes Baleares, Campus UIB, E-07122 Palma de Mallorca, Spain
| | - Pedro V Carelli
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco, Brazil
| | - Tiago Pereira
- Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, São Paulo, Brazil
| | - Fernanda S Matias
- Departamento de Física, Universidade Federal de Alagoas, Maceió, Alagoas, Brazil
| | - Mauro Copelli
- Departamento de Física, Universidade Federal de Pernambuco, Recife, Pernambuco, Brazil
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Kalle P, Sawicki J, Zakharova A, Schöll E. Chimera states and the interplay between initial conditions and non-local coupling. CHAOS (WOODBURY, N.Y.) 2017; 27:033110. [PMID: 28364760 DOI: 10.1063/1.4977866] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Chimera states are complex spatio-temporal patterns that consist of coexisting domains of coherent and incoherent dynamics. We study chimera states in a network of non-locally coupled Stuart-Landau oscillators. We investigate the impact of initial conditions in combination with non-local coupling. Based on an analytical argument, we show how the coupling phase and the coupling strength are linked to the occurrence of chimera states, flipped profiles of the mean phase velocity, and the transition from a phase- to an amplitude-mediated chimera state.
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Affiliation(s)
- Peter Kalle
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Jakub Sawicki
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
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5
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Jaurigue L, Schöll E, Lüdge K. Suppression of Noise-Induced Modulations in Multidelay Systems. PHYSICAL REVIEW LETTERS 2016; 117:154101. [PMID: 27768335 DOI: 10.1103/physrevlett.117.154101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/23/2016] [Indexed: 06/06/2023]
Abstract
Many physical systems involve time-delayed feedback or coupling. In such delay systems, noise can give rise to undesirable oscillations at frequencies resonant to the delay times. We investigate how an additional feedback term can suppress noise-induced modulations in delay systems with self-feedback that exhibit deterministic oscillatory dynamics. A simple characteristic equation is derived to predict optimal delay times for the prototypical example of a Stuart-Landau oscillator subject to two feedback terms. We then show that a characteristic equation of the same form accurately describes the dominant Floquet modes of more complex oscillatory systems and hence can be used to optimize the suppression of noise-induced modulations. This is shown for mode-locked lasers and FitzHugh-Nagumo oscillators subject to self-feedback.
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Affiliation(s)
- Lina Jaurigue
- Institut für Theoretische Physik, Sekretariat EW 7-1, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Sekretariat EW 7-1, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
| | - Kathy Lüdge
- Institut für Theoretische Physik, Sekretariat EW 7-1, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
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6
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Illing L. Amplitude death of identical oscillators in networks with direct coupling. Phys Rev E 2016; 94:022215. [PMID: 27627306 DOI: 10.1103/physreve.94.022215] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2016] [Indexed: 06/06/2023]
Abstract
It is known that amplitude death can occur in networks of coupled identical oscillators if they interact via diffusive time-delayed coupling links. Here we consider networks of oscillators that interact via direct time-delayed coupling links. It is shown analytically that amplitude death is impossible for directly coupled Stuart-Landau oscillators, in contradistinction to the case of diffusive coupling. We demonstrate that amplitude death in the strict sense does become possible in directly coupled networks if the node dynamics is governed by second-order delay differential equations. Finally, we analyze in detail directly coupled nodes whose dynamics are described by first-order delay differential equations and find that, while amplitude death in the strict sense is impossible, other interesting oscillation quenching scenarios exist.
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Affiliation(s)
- Lucas Illing
- Physics Department, Reed College, Portland, Oregon 97202, USA
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Kyrychko YN, Blyuss KB, Schöll E. Synchronization of networks of oscillators with distributed delay coupling. CHAOS (WOODBURY, N.Y.) 2014; 24:043117. [PMID: 25554037 DOI: 10.1063/1.4898771] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
This paper studies the stability of synchronized states in networks, where couplings between nodes are characterized by some distributed time delay, and develops a generalized master stability function approach. Using a generic example of Stuart-Landau oscillators, it is shown how the stability of synchronized solutions in networks with distributed delay coupling can be determined through a semi-analytic computation of Floquet exponents. The analysis of stability of fully synchronized and of cluster or splay states is illustrated for several practically important choices of delay distributions and network topologies.
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Affiliation(s)
- Y N Kyrychko
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
| | - K B Blyuss
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, United Kingdom
| | - E Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
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D'Huys O, Jüngling T, Kinzel W. Stochastic switching in delay-coupled oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032918. [PMID: 25314515 DOI: 10.1103/physreve.90.032918] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/24/2014] [Indexed: 06/04/2023]
Abstract
A delay is known to induce multistability in periodic systems. Under influence of noise, coupled oscillators can switch between coexistent orbits with different frequencies and different oscillation patterns. For coupled phase oscillators we reduce the delay system to a nondelayed Langevin equation, which allows us to analytically compute the distribution of frequencies and their corresponding residence times. The number of stable periodic orbits scales with the roundtrip delay time and coupling strength, but the noisy system visits only a fraction of the orbits, which scales with the square root of the delay time and is independent of the coupling strength. In contrast, the residence time in the different orbits is mainly determined by the coupling strength and the number of oscillators, and only weakly dependent on the coupling delay. Finally we investigate the effect of a detuning between the oscillators. We demonstrate the generality of our results with delay-coupled FitzHugh-Nagumo oscillators.
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Affiliation(s)
- Otti D'Huys
- Institute of Theoretical Physics, University of Würzburg, 97074 Würzburg, Germany
| | - Thomas Jüngling
- Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Wolfgang Kinzel
- Institute of Theoretical Physics, University of Würzburg, 97074 Würzburg, Germany
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Höfener JM, Sethia GC, Gross T. Amplitude death in networks of delay-coupled delay oscillators. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2013; 371:20120462. [PMID: 23960220 DOI: 10.1098/rsta.2012.0462] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
Amplitude death is a dynamical phenomenon in which a network of oscillators settles to a stable state as a result of coupling. Here, we study amplitude death in a generalized model of delay-coupled delay oscillators. We derive analytical results for degree homogeneous networks which show that amplitude death is governed by certain eigenvalues of the network's adjacency matrix. In particular, these results demonstrate that in delay-coupled delay oscillators amplitude death can occur for arbitrarily large coupling strength k. In this limit, we find a region of amplitude death which already occurs at small coupling delays that scale with 1/k. We show numerically that these results remain valid in random networks with heterogeneous degree distribution.
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Affiliation(s)
- Johannes M Höfener
- Biological Physics Section, Max Planck Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, Dresden 01187, Germany
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11
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Kyrychko YN, Blyuss KB, Schöll E. Amplitude and phase dynamics in oscillators with distributed-delay coupling. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2013; 371:20120466. [PMID: 23960224 DOI: 10.1098/rsta.2012.0466] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
This paper studies the effects of distributed-delay coupling on the dynamics in a system of non-identical coupled Stuart-Landau oscillators. For uniform and gamma delay distribution kernels, the conditions for amplitude death are obtained in terms of average frequency, frequency detuning and the parameters of the coupling, including coupling strength and phase, as well as the mean time delay and the width of the delay distribution. To gain further insights into the dynamics inside amplitude death regions, the eigenvalues of the corresponding characteristic equations are computed numerically. Oscillatory dynamics of the system is also investigated, using amplitude and phase representation. Various branches of phase-locked solutions are identified, and their stability is analysed for different types of delay distributions.
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Affiliation(s)
- Y N Kyrychko
- Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, UK.
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Tiana-Alsina J, Hicke K, Porte X, Soriano MC, Torrent MC, Garcia-Ojalvo J, Fischer I. Zero-lag synchronization and bubbling in delay-coupled lasers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:026209. [PMID: 22463301 DOI: 10.1103/physreve.85.026209] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/04/2011] [Indexed: 05/31/2023]
Abstract
We show experimentally that two semiconductor lasers mutually coupled via a passive relay fiber loop exhibit chaos synchronization at zero lag, and study how this synchronized regime is lost as the lasers' pump currents are increased. We characterize the synchronization properties of the system with high temporal resolution in two different chaotic regimes, namely, low-frequency fluctuations and coherence collapse, identifying significant differences between them. In particular, a marked decrease in synchronization quality develops as the lasers enter the coherence collapse regime. Our high-resolution measurements allow us to establish that synchronization loss is associated with bubbling events, the frequency of which increases with increasing pump current.
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Affiliation(s)
- J Tiana-Alsina
- Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Campus de Terrassa, Edif GAIA, Rambla de Sant Nebridi, Terrassa, E-08222 Barcelona, Spain
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13
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Lindley BS, Schwartz IB. Modeling synchronization in networks of delay-coupled fiber ring lasers. OPTICS EXPRESS 2011; 19:24460-24467. [PMID: 22109473 DOI: 10.1364/oe.19.024460] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
We study the onset of synchronization in a network of N delay-coupled stochastic fiber ring lasers with respect to various parameters when the coupling power is weak. In particular, for groups of three or more ring lasers mutually coupled to a central hub laser, we demonstrate a robust tendency toward out-of-phase (achronal) synchronization between the N-1 outer lasers and the single inner laser. In contrast to the achronal synchronization, we find the outer lasers synchronize with zero-lag (isochronal) with respect to each other, thus forming a set of N-1 coherent fiber lasers.
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Affiliation(s)
- Brandon S Lindley
- U.S. Naval Research Laboratory, Code6792, Plasma Physics Division, Nonlinear Systems Dynamics Section, Washington, DC 20375, USA
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Jüngling T, Benner H, Shirahama H, Fukushima K. Complete chaotic synchronization and exclusion of mutual Pyragas control in two delay-coupled Rössler-type oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:056208. [PMID: 22181485 DOI: 10.1103/physreve.84.056208] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2011] [Indexed: 05/31/2023]
Abstract
Two identical chaotic oscillators that are mutually coupled via time delayed signals show very complex patterns of completely synchronized dynamics including stationary states and periodic as well as chaotic oscillations. We have experimentally observed these synchronized states in delay-coupled electronic circuits and have analyzed their stability by numerical simulations and analytical calculations. We found that the conditions for longitudinal and transversal stability largely exclude each other and prevent, e.g., the synchronization of Pyragas-controlled orbits. Most striking is the observation of complete chaotic synchronization for large delay times, which should not be allowed in the given coupling scheme on the background of the actual paradigm.
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Affiliation(s)
- Thomas Jüngling
- Institute for Theoretical Physics, University of Würzburg, Würzburg, Germany.
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Hicke K, D'Huys O, Flunkert V, Schöll E, Danckaert J, Fischer I. Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:056211. [PMID: 21728634 DOI: 10.1103/physreve.83.056211] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/29/2010] [Indexed: 05/31/2023]
Abstract
We study the synchronization properties of the delay dynamics of two identical semiconductor lasers coupled through a semitransparent mirror. Via an analytical and numerical approach, we investigate the influence of asymmetries, in particular mismatches of self- and cross-coupling strength and differences in self- and cross-coupling delay. We show that the former mismatch affects the stability of the zero-lag state but not the dynamics within the synchronization manifold, while the latter mismatch does not affect the quality of synchronization but alters the dynamics significantly. Our results are extended to different unidirectional coupling schemes. This is highly relevant for communication schemes utilizing chaotic dynamics. Finally, the influence of nonlinear gain saturation on the dynamics and stability of synchronization is discussed.
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Affiliation(s)
- K Hicke
- Instituto de Fisica Interdisciplinar y Sistemas Complejos (IFISC), UIB-CSIC, Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
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D'Huys O, Fischer I, Danckaert J, Vicente R. Role of delay for the symmetry in the dynamics of networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:046223. [PMID: 21599286 DOI: 10.1103/physreve.83.046223] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/08/2010] [Revised: 12/21/2010] [Indexed: 05/30/2023]
Abstract
The symmetry in a network of oscillators determines the spatiotemporal patterns of activity that can emerge. We study how a delay in the coupling affects symmetry-breaking and -restoring bifurcations. We are able to draw general conclusions in the limit of long delays. For one class of networks we derive a criterion that predicts that delays have a symmetrizing effect. Moreover, we demonstrate that for any network admitting a steady-state solution, a long delay can solely advance the first bifurcation point as compared to the instantaneous-coupling regime.
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Affiliation(s)
- O D'Huys
- Applied Physics Research Group (APHY), Vrije Universiteit Brussel, 1050 Brussel, Belgium
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Illing L, Hoth G, Shareshian L, May C. Scaling behavior of oscillations arising in delay-coupled optoelectronic oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:026107. [PMID: 21405889 DOI: 10.1103/physreve.83.026107] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2010] [Indexed: 05/30/2023]
Abstract
We study the effect of asymmetric coupling strengths on the onset of oscillations in delay-coupled nonlinear systems. Our experiment consists of two wide-band optoelectronic devices that are cross-coupled. We find that oscillations appear in the system when the product of the two coupling strengths exceeds a critical value. Beyond the oscillation threshold, the oscillation amplitudes grow smoothly and we find a scaling law that describes the dependence of the amplitude on the coupling strengths. The observations are in good agreement with predictions from linear stability analysis and normal form theory.
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Affiliation(s)
- Lucas Illing
- Department of Physics, Reed College, Portland, Oregon 27708, USA.
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